Identifying Nonconvexity in the Sets of Limited-Dimension Quantum Correlations

TL;DR

This paper investigates the geometric properties of quantum correlation sets under dimension constraints, revealing nonconvexities and conditions where quantum correlations can outperform classical distributions.

Contribution

It demonstrates that quantum correlation sets with limited dimension are neither convex nor always containing classical distributions, and explores conditions for convexity and dimensional advantages.

Findings

Quantum correlation sets are nonconvex under certain dimension restrictions.

Quantum correlations can provide advantages over classical distributions in some scenarios.

Convexity of quantum sets depends on the Bell scenario and dimension constraints.

Abstract

Quantum theory is known to be nonlocal in the sense that separated parties can perform measurements on a shared quantum state to obtain correlated probability distributions, which cannot be achieved if the parties share only classical randomness. Here we find that the set of distributions compatible with sharing quantum states subject to some sufficiently restricted dimension is neither convex nor a superset of the classical distributions. We examine the relationship between quantum distributions associated with a dimensional constraint and classical distributions associated with limited shared randomness. We prove that quantum correlations are convex for certain finite dimension in certain Bell scenarios and that they sometimes offer a dimensional advantage in realizing local distributions. We also consider if there exist Bell scenarios where the set of quantum correlations is never convex with finite dimensionality.